Polynomial decay of correlations for flows,
including Lorentz gas examples
Abstract
We prove sharp results on polynomial decay of correlations for nonuniformly hyperbolic flows. Applications include intermittent solenoidal flows and various Lorentz gas models including the infinite horizon Lorentz gas.
1 Introduction
Let be a probability space. Given a measure-preserving flow and observables , we define the correlation function . The flow is mixing if for all .
Of interest is the rate of decay of correlations, or rate of mixing, namely the rate at which converges to zero. Dolgopyat [17] showed that geodesic flows on compact surfaces of negative curvature with volume measure are exponentially mixing for Hölder observables . Liverani [22] extended this result to arbitrary dimensional geodesic flows in negative curvature and more generally to contact Anosov flows. However, exponential mixing remains poorly understood in general.
Dolgopyat [18] considered the weaker notion of rapid mixing (superpolynomial decay of correlations) where for sufficiently regular observables for any fixed , and showed that rapid mixing is ‘prevalent’ for Axiom A flows: it suffices that the flow contains two periodic solutions with periods whose ratio is Diophantine. Field et al. [19] introduced the notion of good asymptotics and used this to prove that amongst Axiom A flows, , an open and dense set of flows is rapid mixing.
In [24], results on rapid mixing were obtained for nonuniformly hyperbolic semiflows, combining the rapid mixing method of Dolgopyat [18] with advances by Young [30, 31] in the discrete time setting. First results on polynomial mixing for nonuniformly hyperbolic semiflows ( for some fixed ) were obtained in [25]. Under certain assumptions the results in [24, 25] were established also for nonuniformly hyperbolic flows. However, for polynomially mixing flows, the assumptions in [25] are overly restrictive and exclude many examples including infinite horizon Lorentz gases.
In this paper, we develop the tools required to cover systematically large classes of nonuniformly hyperbolic flows. The recent review article [26] describes the current state of the art for rapid and polynomial decay of correlations for nonuniformly hyperbolic semiflows and flows and gives a complete self-contained proof in the case of semiflows. Here we provide the arguments required to deal with flows. Our results cover all of the examples in [26].
By [24], rapid mixing holds (at least typically) for nonuniformly hyperbolic flows that are modelled as suspensions over Young towers with exponential tails [30]. See also Remark 8.5. Here we give a different proof that has a number of advantages as discussed in the introduction to [26]. Flows are modelled as suspensions over a uniformly hyperbolic map with an unbounded roof function (rather than as suspensions over a nonuniformly hyperbolic map with a bounded roof function). It then suffices to consider twisted transfer operators with one complex parameter rather than two as in [24], reducing from four to three the number of periodic orbits that need to be considered in Proposition 6.6. Also, the proof of rapid mixing only uses superpolynomial tails for the roof function, whereas [24] requires exponential tails.
Examples covered by our results on rapid mixing include finite Lorentz gases (including those with cusps, corner points, and external forcing), Lorenz attractors, and Hénon-like attractors. We refer to [26] for references and further details.
Examples discussed in [25, 26] for which polynomial mixing holds include nonuniformly hyperbolic flows that are modelled as suspensions over Young towers with polynomial tails [31]. This includes intermittent solenoidal flows, see also Remark 8.6.
The key example of continuous time planar periodic infinite horizon Lorentz gases is considered at length in Section 9. In the finite horizon case, exponential decay of correlations for the flow was proved in [4]. In the infinite horizon case it has been conjectured [20, 23] that the decay rate for the flow is . (An elementary argument in [5] shows that this rate is optimal; the argument is reproduced in the current context in Proposition 9.14.) We obtain the conjectured decay rate for planar infinite horizon Lorentz flows in Theorem 9.1.
Remark 1.1
(a) In [25], the decay rate was proved for infinite horizon Lorentz gases at the semiflow level (after passing to a suspension over a Markov extension and quotienting out stable leaves as in Sections 3 and 6).
It was claimed in [25] that this result held also in certain special cases for the Lorentz flow, and that the decay rate held for all in complete generality. The spurious factor of was then removed in an unpublished preprint “Decay of correlations for flows with unbounded roof function, including the
infinite horizon planar periodic Lorentz gas” by the first and third authors.
Unfortunately these results for flows do not apply to Lorentz gases since hypothesis (P1) in [25] is not satisfied.
The situation is rectified in the current paper. (The unpublished preprint also contained correct results on statistical limit laws such as the central limit theorem for flows with unbounded roof functions. These aspects are completed and extended in [7].)
(b)
A drawback of the method in this paper, already present in [18] and inherited by [24, 25, 26], is that at least one of the observables or is required to be in the flow direction. Here can be estimated, with difficulty, but is likely to be quite large. In the case of the infinite horizon Lorentz gas, this excludes certain physically important observables such as velocity. A reasonable project is to attempt to combine methods in this paper with the methods for (stretched) exponential decay in [4, 12] to obtain the decay rate for Hölder observables and (cf. the second open question in [26, Section 9]).
In Part I of this paper, we consider results on rapid mixing and polynomial mixing for a class of suspension flows over infinite branch uniformly hyperbolic transformations [30]. In Part II, we show how these results apply to important classes of nonuniformly hyperbolic flows including those mentioned in this introduction. The methods of proof in this paper, especially those in Part I, are fairly straightforward adaptations of those in [26]. The main new contribution of the paper (Section 6 together with Part II) is to develop a general framework whereby large classes of nonuniformly hyperbolic flows, including fundamental examples such as the infinite horizon Lorentz gas, are covered by these methods.
Remark 1.2
The paper has been structured to be as self-contained as possible. It does not seem possible to reduce the results on flows in Part I of this paper to the results on semiflows in [26]. Instead, it is necessary to start from scratch and to emulate, rather than apply directly, the methods in [26]. Some of the more basic estimates in [26] are applicable and are collected together at the beginning of Sections 4 (Lemma 4.1 to Proposition 4.9) and Section 5 (Propositions 5.1 to 5.3), as well as in Section 5.2 (Propositions 5.7, 5.11 and 5.12). Also, results on nonexistence of approximate eigenfunctions in [26] are recalled in Sections 6.2 and Section 8.4.
Notation
We use the “big ” and notation interchangeably, writing or if there is a constant such that for all . There are various “universal” constants depending only on the flow that do not change throughout.
Part I Mixing rates for Gibbs-Markov flows
In this part of the paper, we state and prove results on rapid and polynomial mixing for a class of suspension flows that we call Gibbs-Markov flows. These are suspensions over infinite branch uniformly hyperbolic transformations [30]. In Section 2, we recall material on the noninvertible version, Gibbs-Markov semiflows (suspensions over infinite branch uniformly expanding maps). In Section 3, we consider skew product Gibbs-Markov flows where the roof function is constant along stable leaves and state our main theorems for such flows, namely Theorem 3.1 (rapid mixing) and Theorem 3.2 (polynomial mixing). These are proved in Sections 4 and 5 respectively. In Section 6, we consider an enlarged class of Gibbs-Markov flows that can be reduced to skew products and for which Theorems 3.1 and 3.2 remain valid.
We quickly review notation associated with suspension semiflows and suspension flows. Let be a probability space and let be a measure-preserving transformation. Let be an integrable roof function. Define the suspension semiflow/flow
| (1.1) |
where and computed modulo identifications. An -invariant probability measure on is given by .
2 Gibbs-Markov maps and semiflows
In this section, we review definitions and notation from [26, Section 3.1] for a class of Gibbs-Markov semiflows built as suspensions over Gibbs-Markov maps. Standard references for background material on Gibbs-Markov maps are [1, Chapter 4] and [2].
Suppose that is a probability space with an at most countable measurable partition and let be a measure-preserving transformation. For , define where the separation time is the least integer such that and lie in distinct partition elements in . It is assumed that the partition separates trajectories, so if and only if . Then is a metric, called a symbolic metric.
A function is -Lipschitz if is finite. Let be the Banach space of Lipschitz functions with norm .
More generally (and with a slight abuse of notation), we say that a function is piecewise -Lipschitz if is finite for all . If in addition, then we say that is uniformly piecewise -Lipschitz. Note that such a function is bounded on partition elements but need not be bounded on .
Definition 2.1
The map is called a (full branch) Gibbs-Markov map if
-
•
is a measurable bijection for each , and
-
•
The potential function is uniformly piecewise -Lipschitz for some .
Definition 2.2
A suspension semiflow as in (1.1) is called a Gibbs-Markov semiflow if there exist constants , such that is a Gibbs-Markov map, is an integrable roof function with , and
| (2.1) |
(Equivalently, is uniformly piecewise -Lipschitz.) It follows that for all .
For , we define the operators
Definition 2.3
A subset is a finite subsystem of if where is the union of finitely many elements from the partition . (Note that is a full one-sided shift on finitely many symbols.)
We say that has approximate eigenfunctions on if for any , there exist constants , and , and sequences , , with and , such that setting ,
| (2.2) |
Remark 2.4
For brevity, the statement “Assume absence of approximate eigenfunctions” is the assumption that there exists at least one finite subsystem such that does not have approximate eigenfunctions on .
3 Skew product Gibbs-Markov flows
In this section, we recall the notion of skew product Gibbs-Markov flow [26, Section 4.1] and state our main results on mixing for such flows.
Let be a metric space with , and let be a piecewise continuous map with ergodic -invariant probability measure . Let be a cover of by disjoint measurable subsets of called stable leaves. For each , let denote the stable leaf containing . We require that for all .
Let denote the space obtained from after quotienting by , with natural projection . We assume that the quotient map is a Gibbs-Markov map as in Definition 2.1, with partition , separation time , and ergodic invariant probability measure .
Let ; these form a partition of and each is a union of stable leaves. The separation time extends to , setting for .
Next, we require that there is a measurable subset such that for every there is a unique . Let define the associated projection . (Note that can be identified with , but in general .)
We assume that there are constants , such that for all ,
| (3.1) | |||||
| for all . | (3.2) |
Let be an integrable roof function with , and define the suspension flow111Strictly speaking, is not always a flow since need not be invertible. However, is used as a model for various flows, and it is then a flow when is the first return to , so it is convenient to call it a flow. as in (1.1) with ergodic invariant probability measure .
In this subsection, we suppose that is constant along stable leaves and hence projects to a well-defined roof function . It follows that the suspension flow projects to a quotient suspension semiflow . We assume that is a Gibbs-Markov semiflow (Definition 2.2). In particular, increasing if necessary, (2.1) is satisfied in the form
| (3.3) |
We call a skew product Gibbs-Markov flow, and we say that has approximate eigenfunctions if has approximate eigenfunctions (Definition 2.3).
Fix . For , define
(Here denotes absolute value, with regarded as elements of .) Let and be the spaces of observables with and respectively.
We say that is differentiable in the flow direction if the limit exists pointwise. Note that on the set . Define to consist of observables that are -times differentiable in the flow direction with derivatives in , with norm .
We can now state the main theoretical results for skew product Gibbs-Markov flows.
Theorem 3.1
Suppose that is a skew product Gibbs-Markov flow such that for all . Assume absence of approximate eigenfunctions.
Then for any , there exists and such that
Theorem 3.2
Suppose that is a skew product Gibbs-Markov flow such that for some . Assume absence of approximate eigenfunctions. Then there exists and such that
Remark 3.3
Our result on polynomial mixing, Theorem 3.2, implies the result on rapid mixing, Theorem 3.1 (for a slightly more restricted class of observables). However, the proof of Theorem 3.1 plays a crucial role in the proof of Theorem 3.2, justifying the movement of certain contours of integration to the imaginary axis after the truncation step in Section 5.2. Hence, it is not possible to bypass Theorem 3.1 even when only polynomial mixing is of interest.
These results are proved in Sections 4 and 5 respectively. For future reference, we mention the following estimates. Define .
Proposition 3.4
Let . Then
(a) for all , , .
(b) If for some , then .
Proof.
Writing , we compute that
proving part (a). Part (b) is standard (see for example [26, Proposition 8.5]). ∎
4 Rapid mixing for skew product Gibbs-Markov flows
In this section, we consider skew product Gibbs-Markov flows for which the roof function lies in for all . For such flows, we prove Theorem 3.1, namely that absence of approximate eigenfunctions is a sufficient condition for rapid mixing. For notational convenience, we suppose that .
4.1 Some notation and results from [26]
Let and . The Laplace transform of the correlation function is analytic on .
Lemma 4.1 ( [26, Lemma 6.2] )
Let , , . Suppose that
-
(i)
is continuous on and is on for all .
-
(ii)
There exist constants such that
for all , , and all with .
Let . Then there exists a constant depending only on and , such that
∎
Remark 4.2
Since is not a priori well-defined on , the conditions in this lemma should be interpreted in the usual way, namely that extends to a function satisfying the desired conditions (i) and (ii). The conclusion for then follows from a standard uniqueness argument.
For completeness, we provide the uniqueness argument. By [26, Corollary 6.1], the inverse Laplace transform of can be computed by integrating along a contour in . Since on , we can compute the inverse Laplace transform of using the same contour, and we obtain . Hence is well-defined on and satisfies conditions (i) and (ii), so the conclusion follows from [26, Lemma 6.2].
Define and .
Proposition 4.3 ( [26, Proposition 6.3 and Corollary 8.6] )
Let with . Then on where is the Laplace transform of an function for , and
Moreover, .222 All series that we consider on are absolutely convergent for elementary reasons. Details are given in Lemma 4.11 but are generally omitted. ∎
Let denote the transfer operator corresponding to the Gibbs-Markov quotient map . So for all and . Also, for , define the twisted transfer operators
Proposition 4.4
Proof.
This follows from [26, Corollary 7.2]. ∎
For the remainder of this subsection, we suppose that where . Fix with
Let , are as in Section 3. Shrinking if needed, we may suppose without loss that
Let and increase if needed so that .
A function is said to be if is and is -Hölder. Moreover, given and , we write for if for all ,
For and , we write for if in the sense just given for and for , . The same conventions apply to operator-valued functions on .
Remark 4.5
Restricting to as above enables us to obtain estimates for the rapid mixing and polynomially mixing situations simultaneously hence avoiding a certain amount of repetition. The trade off is that the proof of Theorem 3.1 is considerably more difficult. The reader interested only in the rapid mixing case can restrict to integer values of with greatly simplified arguments [26, Section 7] (also see version 3 of our preprint on arxiv).
Following [26, Section 7.4], there exist constants and a scale of equivalent norms
on such that
| (4.1) |
Proposition 4.6
There is a constant such that
Proof.
It is shown in [26, Proposition 8.7] that . Using the definition of , the desired estimate follows by exactly the same argument. ∎
Remark 4.7
Estimates such as those for in Proposition 4.6 hold equally for for all . We use this observation without comment throughout.
Define for . Let . We have the key Dolgopyat estimate:
Proposition 4.8
Assume absence of approximate eigenfunctions. Then is a well-defined bounded operator for . Moreover, for any , there exists such that
∎
Proof.
Proposition 4.9 ( [26, Proposition 7.8 and Corollary 7.9] )
There exists such that has a family of simple eigenvalues , , isolated in , with , , . The corresponding spectral projections form a family of operators on with . ∎
4.2 Approximation of and
The first step is to approximate by functions that are constant on stable leaves and hence well-defined on .
For , define ,
Proposition 4.10
Let . Then
-
(a)
is constant along stable leaves.
-
(b)
.
Proof.
Part (a) is immediate from the definition and part (b) follows by induction. ∎
Also, for , we define ,
Lemma 4.11
Let . Then
on , where
All of these series are absolutely convergent exponentially quickly, pointwise on .
Proof.
Since this result is set in the right-half complex plane, the final statement is elementary. We sketch the arguments. Let with . It is clear that and . Hence and . Similarly, and . As an operator on , we have . Hence .
Also, by Proposition 4.10(b), for each , ,
Next,
Altogether,
where
Now
This completes the proof. ∎
For , we define the approximation operators
for , .
Proposition 4.12
(a) Let , . Then for all , ,
(b) Let , . Then for all , ,
(c) Let , . Then for all , ,
Proof.
We end this subsection by noting for all the identities
4.3 Estimates for and
We continue to suppose that where , and that , , , are as in Subsection 4.1. Let . As shown in the proofs of Propositions 4.14 and 4.15 below, and are Laplace transforms of functions . In this subsection, we obtain estimates for these functions .
Proposition 4.13
There is a constant such that
Proof.
Now
For ,
For ,
Finally for ,
completing the proof. ∎
Proposition 4.14
There is a constant such that
Proof.
Proposition 4.15
There is a constant such that
4.4 Estimates for
For the moment, we suppose that where , and that , , , are as in Subsection 4.1. First, we estimate the inverse Laplace transform associated to .
Proposition 4.16
There is a constant such that
Proposition 4.17
There exists such that
Proof.
For , note that consists of terms (counting repetitions) of the form
where , , , . Since ,
Hence . ∎
Proposition 4.18
Let . Define . Then
Proof.
From now on, we specialize to the rapid mixing case, so and are arbitrarily large and all functions previously regarded as are now . Note that
| (4.2) |
Proposition 4.19
For each there exists such that
Proof.
Fix a -cylinder for the Gibbs-Markov map . Since is a partition element,
| (4.3) |
Define , . Let and be as in Proposition 4.9, and recall that .
Proposition 4.20
For each , there exists such that for all , , and all with ,
-
(a)
for ,
-
(b)
for .
Proof.
For , it follows from Proposition 4.8 that for some . Combining these estimates,
completing the proof of (a).
Next, suppose that . By Proposition 4.9, where is the spectral projection corresponding to and . By Proposition 4.9, is a family of isolated eigenvalues with , and , and is a family of operators on with . Also
where is on . Hence
where is on . Also, , so
where
It follows from the estimates for and that for . Since , the proof of Proposition 4.17 applies equally to , so for .
Finally completing the proof of part (b). ∎
By Lemma 4.11, is analytic on . As shown in the next result, extends smoothly to .
Corollary 4.21
Assume absence of eigenfunctions, and let . There exists such that
for all with , and all with .
Proof.
Proof of Theorem 3.1 Recall that and can be taken arbitrarily large. Hence it follows from Proposition 4.3 that for all . Similarly, by Propositions 4.14 and 4.15, and . Combining these with Corollary 4.21 and substituting into Lemma 4.11, we have shown that extends to . Moreover, we have shown that for every there exists such that
for all with . The result now follows from Lemma 4.1 and Remark 4.2. ∎
5 Polynomial mixing for skew product Gibbs-Markov flows
In this section, we consider skew product Gibbs-Markov flows for which the roof function satisfies for some . For such flows, we prove Theorem 3.2, namely that absence of approximate eigenfunctions is a sufficient condition to obtain the mixing rate .
If is integrable, we write if the inverse Fourier transform of is . We also write instead of for .
Proposition 5.1 ( [26, Proposition 8.2] )
Let be an integrable function such that as . If , then . ∎
The convolution of two integrable functions is defined to be .
Proposition 5.2 ( [26, Proposition 8.4] )
Fix with . Suppose that are integrable and there exist constants such that and for . Then there exists a constant depending only on and such that for . ∎
Proposition 5.3
Define for . Then there exists such that for all .
Proof.
This is contained in the proof of [26, Proposition 8.13]. ∎
5.1 Modified estimate for
Proposition 5.4
There exists such that
for all such that is independent of , and all , .
Proof.
Recall that
Hence . It follows that
| (5.1) |
where .
Let denote the inverse Laplace transform associated to .
Proposition 5.5
There is a constant such that
for all with and all , .
Corollary 5.6
Let be with for all . Then for all , .
5.2 Truncation
We proceed in a manner analogous to [26, Section 8.4], replacing by a bounded roof function. Given , let . Define on and elsewhere. (Unlike [26], it is not sufficient to take .) Note that by (3.3).
Consider the suspension semiflows and on and respectively. (Here, is computed modulo the identification on .) Let and denote the respective correlation functions. In particular, where the observables are the restrictions of to .
Proposition 5.7 ( [26, Proposition 8.19] )
There are constants , such that
for all , , . ∎
We make the following abuse of notation regarding norms of observables . Define where is the extension of by zero to . (In other words, the factor of on the denominator in the definition of is not replaced by .)
With this convention, restricts to with . The similar convention applies to observables . However, restricting to need not preserve smoothness in the flow direction. Below we prove:
Lemma 5.8
Assume absence of approximate eigenfunctions. In particular, there is a finite union of partition elements such that the corresponding finite subsystem does not support approximate eigenfunctions. Choose .
There exist , such that
for all , , , .
Proof of Theorem 3.2 Let , be as in Lemma 5.8. As discussed above, the observable restricts to an observable with no increase in the value of , but restricting to need not preserve smoothness in the flow direction. To circumvent this, following [25, 26] we define an approximating observable , ,
where the are linear combinations of and , , with coefficients independent of and uniquely specified by the requirements and for . 333In fact for but the remaining formulas are messier. When , for instance, , .
It is immediate from the definitions that is -times differentiable in the flow direction. We claim that for some constant independent of . By Lemma 5.8,
Also,
so
Taking , the result follows directly from Proposition 5.7.
It remains to verify the claim. Fix . Let , where lie in the same partition element. Then
where is a constant independent of . Also, by (3.3), for
Hence
This completes the verification of the claim on the region and the other regions are easier to treat. ∎
Our strategy for proving Lemma 5.8 is identical to that for [26, Lemma 8.20]. The first step is to show that the inverse Laplace transform of can be computed using the imaginary axis as the contour of integration.
Proposition 5.9
Let , . Then there exists , , , such that is continuous on and for all with .
Proof.
In this proof, the constant is not required to be uniform in . Consequently, the estimates are very straightforward compared to other estimates in this section.
The desired properties for will hold provided they are verified for all the constituent parts in Lemma 4.11. Note that if is integrable on , then satisfies the required properties with . Hence the estimate in Proposition 4.16 already suffices for . Also, the proof of Proposition 4.19 suffices after truncation since becomes . (Actually, the factor is easily improved to which is integrable when so truncation is not absolutely necessary for the term .)
By definition of , the truncated roof function coincides with on the subsystem , so absence of approximate eigenfunctions passes over to the truncated flow for each . Since , all estimates related to and in Section 4 now hold for arbitrarily large. Hence the arguments in Section 4 yield the desired properties for . Also, it is immediate from the proof of [26, Proposition 6.3] that so for and hence is integrable.
It remains to consider the terms and . Here, we must take into account that the factor of in the definition of is not truncated. Starting from the end of the proof of Proposition 4.14, we obtain
A simplified version of Proposition 4.13 combined with Proposition 3.4(b) yields
Hence . Similarly . Hence and are integrable, completing the proof. ∎
Choose to be and compactly supported such that on a neighbourhood of zero. Let , .
Corollary 5.10
Let , , , . Then
Proof.
From now on we suppress the superscript “” for sake of readability. Notation , and so on refers to the operators obtained using instead of . We end this subsection by recalling some further estimates from [26]. The first is a uniform version of Proposition 4.8.
Proposition 5.11 ( [26, Proposition 8.27] )
Assume absence of approximate eigenfunctions. Then there exists such that
∎
The remaining estimates in this subsection are required when is close to zero. By Proposition 4.9, for each there exists such that
where and are on and , and . In fact, as shown in [26, Section 8.5], can be chosen uniformly in . Moreover, is bounded uniformly in on , so are uniformly in on .
Define
Proposition 5.12
There exists a constant , uniform in , such that
5.3 Proof of Lemma 5.8
Let and be as in Corollary 5.10 with the extra property that . By Proposition 5.1,
| (5.6) |
By Corollary 5.10, we need to show that and for all , , uniformly in .
By Propositions 4.3, 4.14 and 4.15, . (Estimates such as these that hold even before truncation are clearly independent of .) By (5.6) and Proposition 5.2, uniformly in ,
Hence it remains to estimate and . The next lemma provides the desired estimates and completes the proof of Lemma 5.8 (recall that ).
Lemma 5.13
Assume absence of approximate eigenfunctions. There exists , , such that after truncation, uniformly in ,
-
(a)
, and
-
(b)
,
for all , , .
Proof.
(a) Let and recall that
By Proposition 5.11, we can choose such that uniformly in . Write , where is , vanishes in a neighborhood of zero, and is . Then
The estimates for and in Proposition 4.17 and Corollary 5.6 hold even before truncation and hence are uniform in . Using (5.6) and Propositions 5.1 and 5.2,
uniformly in . Since , it follows from Proposition 5.2 that uniformly in ,
Also, by Proposition 4.16 and this is uniform in . Applying Proposition 5.2 once more, uniformly in ,
and part (a) follows.
(b)
As in the proof of Proposition 4.20, we write
where
Here, .
6 General Gibbs-Markov flows
In this section, we assume the setup from Section 3 but we drop the requirement that is constant along stable leaves.
In Subsection 6.1, we introduce a criterion, condition (H), that enables us to reduce to the skew product Gibbs-Markov maps studied in Sections 3, 4 and 5. This leads to an enlarged class of Gibbs-Markov flows for which we can prove results on mixing rates (Theorem 6.4 below). In Subsection 6.2, we recall criteria for absence of approximate eigenfunctions based on periodic data.
6.1 Condition (H)
Let be a map as in Section 3 with quotient Gibbs-Markov map , and define . Let be an integrable roof function with and associated suspension flow .
We no longer assume that is constant along stable leaves. Instead of condition (3.3) we require that
| (6.1) |
(Clearly, if is constant along stable leaves, then conditions (3.3) and (6.1) are identical.)
Recall that is the projection along stable leaves. Define
for all such that the series converges absolutely. We assume
-
(H)
-
(a)
The series converges almost surely on and .
-
(b)
There are constants , such that
-
(a)
When conditions (6.1) and (H) are satisfied, we call a Gibbs-Markov flow. (If is constant along stable leaves then , so every skew product Gibbs-Markov flow is a Gibbs-Markov flow.)
Since , it follows that for all sufficiently large. For simplicity we suppose from now on that (otherwise, replace by ).
Define
| (6.2) |
Note that and , so is an integrable roof function. Hence we can define the suspension flow . Also, a calculation shows that , so is constant along stable leaves and we can define the quotient roof function with quotient semiflow .
In the remainder of this section, we prove that is a skew product Gibbs-Markov flow (and hence is a Gibbs-Markov semiflow), and show that (super)polynomial decay of correlations for is inherited by .
Proposition 6.1
Let be a Gibbs-Markov flow. Then is a skew product Gibbs-Markov flow.
Proof.
Corollary 6.2
There is a constant such that
Proof.
Next, we relate the two suspension flows and . Note that is identified with in the first flow and is identified with in the second flow. Define
computed modulo identifications. Using (6.2) and the identifications on ,
so respects the identification on and hence is well-defined. It follows easily that is a measure-preserving semiconjugacy between the two suspension flows. Similarly, is well-defined and .
Given observables , let . When speaking of and so on, we use the metric on instead of . Let .
Let and where
Lemma 6.3
Let , , for some . Then , , and , .
Proof.
We have . It is immediate that .
Now let . Suppose without loss that . First, we consider the case , . By (H)(b) and the definition of ,
Second, we consider the case . Then we can write , where , so
Proceeding as in the first case,
This leaves the case and . This is impossible since . Hence
so .
The estimate for splits into cases similarly. Let . Then
This leaves the case and . But then , so we obtain . Hence completing the estimate for . The calculation for is similar. ∎
We say that a Gibbs-Markov flow has approximate eigenfunctions if this is the case for (equivalently ).
Theorem 6.4
Suppose that is a Gibbs-Markov flow such that for some . Assume absence of approximate eigenfunctions. Then there exists and such that
6.2 Periodic data and absence of approximate eigenfunctions
In this subsection, we recall the relationship between periodic data and approximate eigenfunctions and review two sufficient conditions to rule out the existence of approximate eigenfunctions. We continue to assume that is a Gibbs-Markov flow as in Subsection 6.1.
Define . Similarly, define and . If is a periodic point of period for (that is, ), then is periodic of period for (that is, ). Recall that is the quotient projection.
Proposition 6.5
Suppose that there exist approximate eigenfunctions on . Let be as in Definition 2.3. If is a periodic point with and where , then
| (6.3) |
The following Diophantine condition is based on [18, Section 13]. (Unlike in [18], we have to consider periods corresponding to three periodic points instead of two.)
Proposition 6.6
Let be fixed points for , and let , , be the corresponding periods for . Let be the finite subsystem corresponding to the three partition elements containing .
If is Diophantine, then there do not exist approximate eigenfunctions on .
The condition in Proposition 6.6 is satisfied with probability one but is not robust. Using the notion of good asymptotics [19], we obtain an open and dense condition.
Proposition 6.7
Let be a finite subsystem. Let be a fixed point for with period for the flow. Let , , be a sequence of periodic points with such that their periods for the flow satisfy
where , are constants, is a bounded sequence with , and either (i) and , or (ii) and for some . (Such a sequence of periodic points is said to have good asymptotics.)
Then there do not exist approximate eigenfunctions on .
By [19], for any finite subsystem , the existence of periodic points with good asymptotics in is a -open and -dense condition. Although [19] is set in the uniformly hyperbolic setting, the construction applies directly to the current set up as we now explain. Assume that is a Riemannian manifold. Let and be two of the partition elements in and set for . Assume that , are submanifolds of and that and are when restricted to for some .
Let be a fixed point for and choose a transverse homoclinic point in . Following [19], we construct a sequence of -periodic points , , for with orbits lying in . The sequence automatically has good asymptotics except that in exceptional cases it may be that . By [19], the liminf is positive for a open and dense set of roof functions .
Combining this construction with Proposition 6.7, it follows that nonexistence of approximate eigenfunctions holds for an open and dense set of smooth Gibbs-Markov flows.
Part II Mixing rates for nonuniformly hyperbolic flows
In this part of the paper, we show how the results for suspension flows in Part I can be translated into results for nonuniformly hyperbolic flows defined on an ambient manifold. In Section 7, we show how this is done under the assumption that condition (H) from Section 6 is valid. In Section 8, we describe a number of situations where condition (H) is satisfied. This includes all the examples considered here and in [26]. In Section 9, we consider in detail the planar infinite horizon Lorentz gas.
7 Nonuniformly hyperbolic flows and suspension flows
In this section, we describe a class of nonuniformly hyperbolic flows that have most of the properties required for to be modelled by a Gibbs-Markov flow. (The remaining property, condition (H) from Section 6, is considered in Section 8.)
In Subsection 7.1, we consider a class of nonuniformly hyperbolic transformations modelled by a Young tower [30, 31], making explicit the conditions from [30] that are needed for this paper. In Subsection 7.2, we consider flows that are Hölder suspensions over such a map and show how to model them, subject to condition (H), by a Gibbs-Markov flow. In Subsection 7.3, we generalise the Hölder structures in Subsection 7.2 to ones that are dynamically Hölder.
In applications, is typically a first-hit Poincaré map for the flow and hence is invertible. Invertibility is used in Proposition A.1 but not elsewhere, so many of our results do not rely on injectivity of .
7.1 Nonuniformly hyperbolic transformations
Let be a measurable transformation defined on a metric space with . We suppose that is nonuniformly hyperbolic in the sense that it is modelled by a Young tower [30, 31]. We recall the metric parts of the theory; the differential geometry part leading to an SRB or physical measure does not play an important role here.
Product structure
Let be a measurable subset of . Let be a collection of disjoint measurable subsets of (called “stable leaves”) and let be a collection of disjoint measurable subsets of (called “unstable leaves”) such that each collection covers . Given , let and denote the stable and unstable leaves containing .
We assume that for all , the intersection consists of precisely one point, denoted , and that . Also we suppose there is a constant such that
| (7.1) |
Induced map
Next, let be an at most countable measurable partition of such that for all . Also, fix constant on partition elements such that for all . Define by . Let be an ergodic -invariant probability measure on and suppose that is integrable. (It is not assumed that is the first return time to .)
Contraction/expansion
Let ; these form a partition of and each is a union of stable leaves. The separation time extends to , setting for .
We assume that there are constants , such that for all , ,
| (7.2) | |||||
| (7.3) |
where is the number of returns of to by time . Note that conditions (3.1) and (3.2) are special cases of (7.2) and (7.3) where can be chosen to be any fixed unstable leaf. In particular, all the conditions on in Sections 3 and 6 are satisfied.
Remark 7.1
Further hypotheses in [30] ensure the existence of SRB measures on , and . These assumptions are not required here and no special properties of and (other than the properties mentioned above) are used.
Remark 7.2
The abstract setup in [30] essentially satisfies all of the assumptions above. However condition (7.2) is stated in the slightly weaker form . As pointed out in [16], the stronger form (7.2) is satisfied in all known examples where the weaker form holds.
Condition (7.4) is not stated explicitly in [30] but is an automatic consequence of the set up therein provided is injective. We provide the details in Proposition A.1. In the examples considered in this paper and in [26], the map is a first return map for a flow and hence is injective, so condition (7.4) is not very restrictive.
Condition (7.4) is also used in [26, Section 5.2] but is stated there in a slightly different form. In [26], the subspace is not needed (and hence not mentioned) and the stable and unstable disks , are replaced by their intersections with . Hence the condition for in [26, Section 5.2] becomes for in our present notation and hence holds by (7.4).
Proposition 7.3
for all , .
Proof.
Let . Note that and . Hence
as required. ∎
7.2 Hölder flows and observables
Let be a flow defined on a metric space with . Fix .
Given , define and . Let . Also, define and let . (Such observables are Hölder in the flow direction.)
We say that is differentiable in the flow direction if the limit exists pointwise. Define and let .
Let be a Borel subset and define using the metric restricted to . We suppose that for all , where lies in and . In addition, we suppose that for any there exists such that
| (7.5) |
Define by . We suppose that is a nonuniformly hyperbolic transformation as in Subsection 7.1, with induced map and so on.
Define . We define the induced roof function
Note that so and . Define the suspension flow as in (1.1).
To deduce rates of mixing for nonuniformly hyperbolic flows from the corresponding result for Gibbs-Markov flows, Theorem 6.4, we need to verify that
-
(i)
Condition (6.1) holds.
-
(ii)
Condition (H) from Section 6 holds.
-
(iii)
Regular observables on lift to regular observables on .
Ingredients (i) and (ii) guarantee that the suspension flow is a Gibbs-Markov flow and ingredient (iii) ensures that Theorem 6.4 applies to the appropriate observables on .
In the remainder of this subsection, we deal with ingredients (i) and (iii). First, we verify that satisfies condition (6.1). Let and .
Proposition 7.4
Let for some and let . Then
Moreover,
Proof.
Next we deal with ingredient (iii) assuming (ii). Define as .
Proposition 7.5
Suppose that the function satisfies condition (H).
Then observables lift to observables that lie in where and the metric on is replaced by the metric .
For , observables lift to observables .
Moreover, there is a constant such that and .
Proof.
Let . We show that . The same calculation with shows that , so , We take with corresponding value of in (7.5)
Let with for some . There exists such that
Suppose without loss that . Then
where and . Note that . Hence and so . Similarly, and . Also, . By (7.5) and (7.6),
Since , it follows from Proposition 7.4 that
Hence
whenever . For , we have the estimate , so in all cases we obtain
Also,
so as required. ∎
7.3 Dynamically Hölder flows and observables
The Hölder assumptions in Subsection 7.2 can be replaced by dynamically Hölder as follows. We continue to assume that .
Definition 7.6
The roof function , the flow and the observable are dynamically Hölder if for some and there is a constant such that for all , ,
-
(a)
for all .
-
(b)
For every , there exist such that , and setting ,
Also, we replace the assumption by the condition that lies in and satisfies (b) for all .
Remark 7.7
It is easily verified that condition (6.1) remains valid under the more relaxed assumption on in Definition 7.6(a). Also, it follows as in the proof of Proposition 7.5 that .
Next we estimate and for , where . If , then as in the proof of Proposition 7.5. Hence we can suppose that for some . Set and choose as in Definition 7.6(b). Then
Hence for all , and so .
To proceed, we recall that , so by (7.4). Hence
| (7.7) |
To control , we assume without loss that , and distinguish three cases.
If , we argue as in the bound for .
If , then there exists and such that and . By Corollary 6.2 and (7.7),
and so
On the other hand, choosing and for as in Definition 7.6(b), we get
where we have used (7.7) and . Hence
This completes the verification that . A similar argument shows that , completing the verification that Proposition 7.5 holds under the modified assumptions.
8 Condition (H) for nonuniformly hyperbolic flows
In this section, we consider various classes of nonuniformly hyperbolic flows for which condition (H) in Section 6 can be satisfied. We are then able to apply Theorem 6.4 to obtain results that superpolynomial and polynomial mixing applies to such flows as follows:
Corollary 8.1
Let be a nonuniformly hyperbolic flow as in Section 7.2 and assume that condition (H) is satisfied. Then
(a) is a Gibbs-Markov flow.
(b) Suppose that for some and assume absence of approximate eigenfunctions for . Then there exists and such that
for all , , .
Proof.
Part (a) follows from the discussion in Section 7.2 (so ingredient (i) is automatic and ingredient (ii) is now assumed).
The analogous result holds for nonuniformly hyperbolic flows and observables satisfying the dynamically Hölder conditions in Section 7.3.
We verify condition (H) for three classes of flows. In Subsection 8.1, we consider roof functions with bounded Hölder constants. In Subsection 8.2, we consider flows for which there is exponential contraction along stable leaves. In Subsection 8.3, we consider flows with an invariant Hölder stable foliation. These correspond to the situations mentioned in [26, Section 4.2].
Also, in Subsection 8.4, we briefly review the temporal distance function and a criterion for absence of approximate eigenfunctions.
8.1 Roof functions with bounded Hölder constants
We assume a “bounded Hölder constants” condition on , namely that for all ,
| for all , | (8.1) | ||||
| for all , . | (8.2) |
This leads directly to an enhanced version of (6.1):
Proposition 8.2
for all , .
Proof.
Let . Then
as required. ∎
8.2 Exponential contraction along stable leaves
In this subsection, we suppose that and that is exponentially contracting along stable leaves:
| (8.3) |
Note that this strengthens condition (7.2). Proposition 7.3 becomes
| (8.4) |
Lemma 8.4
If condition (8.3) is satisfied, then condition (H) holds.
Proof.
Let , . We verify condition (H) with and , using the equivalent definition for ,
Next, let and set . Write , where
By the calculation for , we obtain for all . Also, , so for .
Remark 8.5
In cases where lies in and the dynamics on is modelled by a Young tower with exponential tails (so for some ), it is immediate that for all and that condition (8.3) is satisfied. Assuming absence of approximate eigenfunctions, we obtain rapid mixing for such flows.
8.3 Flows with an invariant Hölder stable foliation
Let be a Hölder nonuniformly hyperbolic flow as in Section 7.2. For simplicity, we suppose that is a Riemannian manifold and that is a smoothly embedded cross-section for the flow. We assume that the flow possesses a -invariant Hölder stable foliation in a neighbourhood of . (A sufficient condition for this to hold is that is a partially hyperbolic attracting set with a -invariant dominated splitting , see [3].) We also assume that can be chosen arbitrarily small. In this subsection, we show how to use the stable foliation for the flow to show that is Hölder, hence verifying the hypotheses in Section 6.1.
Remark 8.6
First, we show that if and coincide for all , then is already a skew product (so ).
Proposition 8.7
Suppose that and coincide for all . Then is constant along stable leaves , .
Proof.
For ,
But setting ,
Hence . ∎
Let for some fixed and define the new cross-section to the flow . Shrinking if necessary, there exists a unique continuous function with such that and . Moreover, is Hölder since is smoothly embedded in and is Hölder by the assumption on the regularity of the stable foliation . Define the new roof function
We observe that is the return time for the flow to the cross-section .
Lemma 8.8
Under the above assumption on , condition (H) holds.
8.4 Temporal distance function
Dolgopyat [18, Appendix] showed that for Axiom A flows a sufficient condition for absence of approximate eigenfunctions is that the range of the temporal distance function has positive lower box dimension. This was extended to nonuniformly hyperbolic flows in [25, 26]. Here we recall the main definitions and result.
We assume that condition (H) holds, so that the suspension flow is a Gibbs-Markov flow (and hence conjugate to a skew product flow). We also assume the dynamically Hölder setup from Section 7.3. In particular, the Poincaré map is nonuniformly hyperbolic as in Section 7.1 and has a local product structure. Also we assume that the roof function has bounded Hölder constants along unstable leaves, so condition (8.2) is satisfied.
Let and set , . Define the temporal distance function ,
It follows from the construction in [26, Section 5.3] (which uses (7.4) and (8.2)) that inverse branches for can be chosen so that is well-defined.
Lemma 8.9 ( [26, Theorem 5.6])
Let where is a union of finitely many elements of the partition . Let denote the corresponding finite subsystem of . If the lower box dimension of is positive, then there do not exist approximate eigenfunctions on . ∎
Remark 8.10
For Axiom A attractors, can be taken to be connected and is continuous, so absence of approximate eigenfunctions is ensured whenever is not identically zero. For nonuniformly hyperbolic flows, where the partition is countably infinite, is a Cantor set of positive Hausdorff dimension [25, Example 5.7]. In general it is not clear how to use this property since is generally at best Hölder. However for flows with a contact structure, a formula for in [21, Lemma 3.2] can be exploited and the lower box dimension of is indeed positive, see [25, Example 5.7]. The arguments in [25, Example 5.7] apply to general Gibbs-Markov flows with a contact structure. A special case of this is the Lorentz gas examples considered in Section 9.
9 Billiard flows associated to infinite horizon Lorentz gases
In this section we show that billiard flows associated to planar infinite horizon Lorentz gases satisfy the assumptions of Section 8.1. In particular, we prove decay of correlations with decay rate .
Background material on infinite horizon Lorentz gases is recalled in Subsection 9.1 and the decay rate is proved in Subsection 9.2. In Subsection 9.3, we show that the same decay rate holds for semidispersing Lorentz flows and stadia. In Subsection 9.4, we show that the decay rate is optimal for the examples considered in this section.
9.1 Background on the infinite horizon Lorentz gas
We begin by recalling some background on billiard flows; for further details we refer to the monograph [13].
Let denote the two dimensional flat torus, and let us fix finitely many disjoint convex scatterers with boundaries of nonvanishing curvature. The complement is the billiard domain, and the billiard dynamics are that of a point particle that performs uniform motion with unit speed inside , and specular reflections — angle of reflection equals angle of incidence — off the scatterers, that is, at the boundary . The resulting billiard flow is , where the phase space is a Riemannian manifold, and preserves the (normalized) Lebesgue measure (often called Liouville measure in the literature).
There is a natural Poincaré section corresponding to collisions (with outgoing velocities), which gives rise to the billiard map denoted by , with absolute continuous invariant probability measure . The time until the next collision, the free flight function , is defined to be . The Lorentz gas has finite horizon if and infinite horizon if is unbounded.
In the finite horizon case, [4] recently proved exponential decay of correlations. In this section, we prove
Theorem 9.1
Let . In the infinite horizon case, there exists such that for all and (and more generally for the class of observables defined in Corollary 9.6 below).
Let us fix some terminology and notations. The billiard map is discontinuous, with singularity set corresponding to the preimages of grazing collisions. Here, is the closure of a countable union of smooth curves, consists of countably many connected components , , and is . If for some , then, in particular, and lie on the same scatterer (even when the configuration is unfolded to the plane). Throughout our exposition, denotes the Euclidean distance of the two points, i.e. the distance that is generated by the Riemannian metric on (or ).
It follows from geometric considerations in the infinite horizon case that . Moreover, as the trajectories are straight lines, we have
| (9.1) | ||||
| (9.2) |
The billiard maps considered here (both finite and infinite horizon) have uniform contraction and expansion even for . There exist stable and unstable manifolds of positive length for almost every , which we denote by and respectively, and there exist constants , such that for all , ,
| (9.3) | ||||
| (9.4) |
This follows from the uniform hyperbolicity properties of , see in particular [13, Formula (4.19)].
Furthermore, there is a constant such that for ,
| (9.5) | ||||
| (9.6) |
To verify (9.5), note that consists of a position and a velocity component. In course of the free flight, the velocities do not change, while for , the position component can only shrink as stable manifolds correspond to converging wavefronts. A similar argument applies to (9.6).
Remark 9.2
(a) In the remainder of the section – and in particular in the proof of Proposition 9.5 below – we apply (9.1) repeatedly, but always in the case when either , or . As all iterates are smooth on local stable manifolds (while all iterates are smooth on local unstable manifolds), both of these conditions imply for some .
(b)
For larger values of than those in (9.5), we note that may grow large temporarily: it can happen that one of the trajectories has already collided with some scatterer, while the other has not, hence even though the two points are close in position, the velocities differ substantially.
Similar comments apply to (9.6). This phenomenon is the main reason why we require the notion of dynamically Hölder flows in Definition 7.6.
In [30], Young constructs a subset and an induced map that possesses the properties discussed in Section 7.1 including (7.4). The tails of the return time are exponential, i.e. for some . Moreover, the construction can be carried out so that is as small as desired. This is proved in [30] for the finite horizon, and in [11] for the infinite horizon case. We mention that (7.2) and (7.3) follow from (9.3) and (9.4), respectively, while (7.1) holds as the stable and the unstable manifolds are uniformly transversal, see [13, Formulas (4.13) and (4.21)].
Proposition 9.3
For all , , and all ,
Proof.
Define the induced roof function . Using (7.3), it is immediate from Proposition 9.3 that has bounded Hölder constants in the sense of Section 8.1:
Proposition 9.5
For sufficiently small, there exist an integer and a constant such that for all , , and all , there exist such that
where .
Proof.
Define for , . By Proposition 9.3, there is a constant such that
| (9.7) |
for all , (which is equivalent to ) and all .
Now consider with , and . Let .
Choosing . By (7.1), . Also, . We can shrink if necessary so that .
Write where and . (When , we take , .) Similarly, write . Note that .
If , then we take where . Then and .
Note that , hence by (9.7). In particular, . Also . Hence by (9.3) and (9.5),
The argument for is analogous.
Choosing . This goes along similar lines. We can shrink and increase so that . Note that .
Since , it follows from (7.4) that . Write where and . Similarly write . Note that .
If , then we take . It follows from (7.3) and (9.7) that
Without loss, , so by (7.3), (9.2) and (9.6),
If , then we take where . Then and .
Corollary 9.6
Let , such that , for all . Suppose also that there is a constant such that and for all of the form , where for some , , , and for all . Then , , and are dynamically Hölder in the sense of Definition 7.6.
9.2 Tail estimate for and completion of the proof of Theorem 9.1
Since
| (9.8) | ||||
| (9.9) |
a standard argument shows that . In fact, we have
Proposition 9.7
.
Lemma 9.8 ( [29, Lemma 16], [15, Lemma 5.1] )
There are constants with the following property. Define
Then for any sufficiently large there is a constant such that
∎
For , define
Corollary 9.9
For sufficiently large, .
Proof.
Fix and as in Lemma 9.8. Also fix sufficiently large.
Let . Define and choose such that . Define . Then and are the two largest free flights during the iterates .
We begin by showing that these two flight times have comparable length. Indeed, let , . Then . Hence
| (9.10) |
In particular, and for large .
Proof of Proposition 9.7 Define the tower with probability measure where . Recall that where .
Finally, by (9.9), for any and so as required. ∎
It follows from Lemma 8.3 and Corollary 9.4 that condition (H) is satisfied. Hence by Corollary 8.1(a), the suspension flow is a Gibbs-Markov flow as defined in Section 6. By Proposition 9.7, . By Corollary 9.6, the flows and observables are dynamically Hölder (Definition 7.6). Hence it follows from Corollary 8.1(b) that absence of approximate eigenfunctions implies decay rate .
Finally, we exclude approximate eigenfunctions. By Corollary 9.4, condition (8.2) holds and hence the temporal distortion function is defined as in Section 8.4. Let be a finite subsystem and let . The presence of a contact structure implies by Remark 8.10 that the lower box dimension of is positive. Hence absence of approximate eigenfunctions follows from Lemma 8.9.
9.3 Semi-dispersing Lorentz flows and stadia
In this subsection we discuss two further classes of billiard flows and show that the scheme presented above can be adapted to cover these examples, resulting in Theorem 9.13.
Semi-dispersing Lorentz flows are billiard flows in the planar domain obtained as where is a rectangle and the are finitely many disjoint convex scatterers with boundaries of nonvanishing curvature. By the unfolding process – tiling the plane with identical copies of , and reflecting the scatterers across the sides of all these rectangles – an infinite periodic configuration is obtained, which can be regarded as an infinite horizon Lorentz gas.
Bunimovich stadia are convex billiard domains enclosed by two semicircular arcs (of equal radii) connected by two parallel line segments. An unfolding process could reduce the bounces on the parallel line segments to long flights in an unbounded domain, however, there is another quasi-integrable effect here corresponding to sequences of consecutive collisions on the same semi-circular arc.
Both of these examples have been extensively studied in the literature, see for instance [9, 13, 14, 25, 6], and references therein. A common feature of the two examples is that the billiard map itself is not uniformly hyperbolic; however, there is a geometrically defined first return map which has uniform expansion rates. As before, the billiard domain is denoted by , and the billiard flow is where . However, this time we prefer to denote the natural Poincaré section by , the corresponding billiard map as , and the free flight function as where . Then, as mentioned above, there is a subset such that the first return map of to has good hyperbolic properties. We denote this first return map by . The corresponding free flight function is given by . Let us, furthermore, introduce the discrete return time given by .
In the case of the semi-dispersing Lorentz flow, corresponds to collisions on the scatterers . In the case of the stadium, corresponds to first bounces on semi-circular arcs, that is, if is on one of the semi-circular arcs, but is on another boundary component (on the other semi-circular arc, or on one of the line segments).
The following properties hold. Unless otherwise stated, standard references are [13, Chapter 8] and [14]. As in section 9.1, always denotes the Euclidean distance of the two points, generated by the Riemannian metric.
-
•
There is a countable partition such that is and is constant for any . We refer to the partition elements with as cells; these are of two different types:
-
–
Bouncing cells are present both in the semi-dispersing billiard examples and in stadia. For these, one iteration of consists of several consecutive reflections on the flat boundary components, that is, the line segments. By the above mentioned unfolding process, these reflections reduce to trajectories along straight lines in the associated unbounded table.
-
–
Sliding cells are present only in stadia. For these, one iteration of consists of several consecutive collisions on the same semi-circular arc.
-
–
-
•
, and , however, there is no uniform upper bound on , and no uniform lower bound for .
- •
-
•
If where is a bouncing cell, in the associated unfolded table the flow trajectories until the first return to are straight lines, hence (9.1) follows. If and is a sliding cell, the induced roof function is uniformly Hölder continuous with exponent , as established in the proof of [6, Theorem 3.1]. The same geometric reasoning applies to as long as . Summarizing, we have
(9.11) for and . In particular, .
-
•
(9.2) has to be relaxed to
(9.12) -
•
(9.5) has to be relaxed to the following two formulas:
(9.13) (9.14) Similarly, (9.6) has to be relaxed to
(9.15) (9.16) To verify (9.16), let us note first that consists of a position and a velocity component, and in course of a free flight velocities do not change. Now the mechanism of hyperbolicity for stadia is defocusing, see, for instance, [13, Figure 8.1], which guarantees that for , the position component of in course of the free flight is dominated by the position component at the end of the free flight. (9.14) holds for analogous reasons. To verify (• ‣ 9.3), by uniform hyperbolicity of (in particular Formula (9.4), see above), it is enough to consider how evolves unstable vectors between two consecutive applications of , ie. within a series of sliding or bouncing collisions. On the one hand, again by the defocusing mechanism, does not contract the p-length of unstable vectors, see [13, Section 8.2]. On the other hand, for an unstable vector, the ratio of the Euclidean and the p-length is , where is the slope of the unstable vector in the standard billiard coordinates, and is the collision angle, see [13, Formula (8.21)]. Now is uniformly bounded away from , see Formula [13, Formula (8.18)], while is constant in course of a sequence of consecutive sliding or bouncing collisions. (9.13) holds by an analogous argument.
-
•
The map can be modeled by a Young tower with exponential tails. In particular, there exists a subset and an induced map that possesses the properties discussed in Section 7.1 including (7.4). The tails of the return time are exponential, i.e. for some .444It is important to note that here is the return time to in terms of ; the return time in terms of has polynomial tails. Moreover, the construction can be carried out so that is as small as desired. The existence of the Young tower satisfying these properties is established in [14]. As in subsection 9.1, we introduce the induced roof function .
-
•
By construction, for , and fixed, and always belong to the same cell of .
Let us introduce and . The following version of Proposition 9.3 holds.
Proposition 9.10
For all , , and all ,
This readily implies
The adapted version of Proposition 9.5 reads as follows.
Proposition 9.12
For sufficiently small, there exist an integer and a constant such that for all , , and all , there exist such that
where .
Proof.
Fix for some , and . We will focus on choosing the appropriate and obtaining the relevant estimates. The choice of is analogous. Recall the notation and note that .
First adjustment. As in the proof of Proposition 9.5, we arrive at and for the same , and such that and . Indeed, a priori we have and , where, as , shrinking if needed, (9.17) implies . If , then let , , and follows from (9.17). If , then , where . Note that , hence . Let , so that and as . Note that we do not claim anything about at this point.
Second adjustment. For brevity, introduce and . We have
for some (note that ), and . Note that by (9.13), (9.14) and (9.3), for any , we have
| (9.18) |
where we have used (9.11). We distinguish three cases: , and .
If , (9.18) along with implies . But then, again by (9.18), (9.13) and (9.14), we have
As , we can fix .
If , we prefer to represent our points as
for some . Now by (9.18) and as , we have . Define
Then , where and
Hence
where by (9.12), while by (9.13), (9.14) and (9.18). Hence , as desired. On the other hand , and as we have already controlled , we have .
The case when can be treated analogously. The choice of goes along similar lines, so we omit the details. ∎
Theorem 9.13
Consider a semi-dispersing Lorentz flow or the billiard flow in a Bunimovich stadium. Let . There exists such that for all and (and more generally for the class of observables defined in Corollary 9.6).
Proof.
It follows from Lemma 8.3 and Corollary 9.11 that condition (H) is satisfied. Hence by Corollary 8.1(a), the suspension flow is a Gibbs-Markov flow as defined in Section 6. The conclusions of Corollary 9.6 follow from Propositions 9.10 and 9.12. Hence the flows and observables are dynamically Hölder (Definition 7.6).
For the tail estimate on , introduce , . Note that , and . Also it is shown in [15] (both for the semi-dispersing examples and for stadia) that . Hence .
9.4 Lower bounds
In this subsection, we show that it is impossible to improve on the error rate for infinite horizon Lorentz gases, semidispersing Lorentz flows, and Bunimovich stadia. The following result is based on [5, Corollary 1.3].
Proposition 9.14
Let with . Suppose that . Then .
Proof.
Let . Then
By the assumption on , we obtain . ∎
In the case of the planar infinite horizon Lorentz gas, Szász & Varjú [29] showed that converges in distribution to a nondegenerate normal distribution for typical Hölder mean zero observables . The result applies equally to semidispersing Lorentz flows. Similarly, in the case of Bunimovich stadia by Bálint & Gouëzel [5, Corollary 1.6]. In particular, . Hence by Proposition 9.14, an upper bound of the type is impossible and so the upper bound in Theorems 9.1 and 9.13 is optimal.
Remark 9.15
There is also the possibility of obtaining an asymptotic expression of the form
| (9.19) |
( arbitrarily small, ) for certain classes of observables . Such results are obtained in [27] in cases where there is a first return to a uniformly hyperbolic map . The first return map in the examples considered here is nonuniformly hyperbolic, modelled by a Young tower with exponential tails, so [27] does not apply directly. In a recent preprint, [10] have announced the existence of a uniformly hyperbolic first return. This combined with [27] may yield the asymptotic (9.19). (Interestingly, the class of observables in (9.19) would be disjoint from the class of observables covered by Proposition 9.14.)
Appendix A Condition (7.4)
In this appendix, we verify that condition (7.4) holds in the abstract framework of [30]. For this purpose, we switch to the notation of [30].
Proposition A.1
Let be an injective transformation satisfying the abstract set up in [30, Section 1]: specifically (P1), the second part of (P2), property (iii) of the separation time , and (P4)(a).
Let , . Then .
Proof.
It follows from injectivity of and hence , as well as (P2), that
| (A.1) |
Recall from (P1) that we have the local product structure . By (P2), is a -subset of which means that for some subset . Hence for all . Also, for all .
Now, (it contains ) so it follows from the above considerations that . Combining this with (A.1),
| (A.2) |
It remains to prove the reverse inclusion, so suppose that . By (P1), there exists . By (A.2), for some .
Since and lie in the same stable disk it follows from property (iii) of the separation time that . Using property (iii) once more, . But so (P4)(a) implies that . Hence . This shows that completing the proof. ∎
Acknowledgements
The research of PB was supported in part by Hungarian National Foundation for Scientific Research (NKFIH OTKA) grants K104745 and K123782. OB was supported in part by EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS). The research of IM was supported in part by a European Advanced Grant StochExtHomog (ERC AdG 320977).
We are grateful to the referees for very helpful comments which led to many clarifications and corrections.
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